$0.5737373...... = $

If  ${x_r} = \cos (\pi /{3^r}) - i\sin (\pi /{3^r}),$ (where  $i = \sqrt{-1}),$ then value  of $x_1.x_2.x_3......\infty ,$ is :-

The sum of $3$ numbers in geometric progression is $38$ and their product is $1728$. The middle number is

If the sum of first 6 term is $9$ times to the sum of first $3$ terms of the same $G.P.$, then the common ratio of the series will be

If the $4^{\text {th }}, 10^{\text {th }}$ and $16^{\text {th }}$ terms of a $G.P.$ are $x, y$ and $z,$ respectively. Prove that $x,$ $y, z$ are in $G.P.$

If $\frac{{x + y}}{2},\;y,\;\frac{{y + z}}{2}$ are in $H.P.$, then $x,\;y,\;z$ are in